Embossed paper is where there are raised patterns on the paper instead of it being smooth.
So the different embossed papers have different patterns on them.

Sand embossed is probably small raised random bumps so it looks like sand is on it.
Embossed linen is suppose to look like cloth with parallel lines.
Embossed cross-hatch looks like criss-cross stitching.

Ivory is suppose to be super white color, but with no other description, it’s probably just flat paper.
Synthetic paper is made with synthetic poly fibers instead of natural wood fibers. Synthetic paper is more durable.

Of these options, the embossed linen probably looks best for a wedding invitation.

My advice since paper manufacturers vary on the different embossed patterns, is to ask for a paper sample. All paper manufacturers should have them and usually send them out free. That way you can actually see and feel what the paper stock looks like.